Early Quantum Theory and Models of the Atom

P R I NC I PLES WITH APPL I CATIONS

DOUGLAS C. GIANCOLI

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Electron microscopes (EM) produce images using electrons which have wave properties just as light does. Because the wavelength of electrons can be much smaller than that of visible light, much greater resolution and magnification can be obtained. A scanning electron microscope (SEM) can produce images with a three-dimensional quality.

All EM images are monochromatic (black and white). Artistic coloring has been added here, as is common. On the left is an SEM image of a blood clot forming (yellow-color web) due to a wound. White blood cells are colored green here for visibility. On the right, red blood cells in a small artery. A red blood cell travels about

15 km a day inside our bodies and lives roughly 4 months before damage or rupture. Humans contain 4 to 6 liters of blood, and 2 to

3 * 10 13 red blood cells.

Early Quantum Theory C 27

and Models of the Atom

CHAPTER-OPENING QUESTION —Guess now!

CONTENTS It has been found experimentally that

27–1 Discovery and Properties

(a) light behaves as a wave.

of the Electron

(b) light behaves as a particle.

27–2 Blackbody Radiation;

(c) electrons behave as particles.

Planck’s Quantum Hypothesis

(d) electrons behave as waves.

27–3 Photon Theory of Light and

(e) the Photoelectric Effect all of the above are true. (f) only (a) and (b) are true.

27–4 Energy, Mass, and Momentum of a Photon

(g) only (a) and (c) are true.

*27–5 Compton Effect

(h) none of the above are true.

27–6 Photon Interactions; Pair Production

T Einstein’s theory of relativity). Unlike the special theory of relativity, the 27–8 Wave Nature of Matter

he second aspect of the revolution that shook the world of physics in the

27–7 Wave–Particle Duality; the

early part of the twentieth century was the quantum theory (the other was

Principle of Complementarity

revolution of quantum theory required almost three decades to unfold, and many

27–9 Electron Microscopes

scientists contributed to its development. It began in 1900 with Planck’s quantum

27–10 Early Models of the Atom

hypothesis, and culminated in the mid-1920s with the theory of quantum

27–11 Atomic Spectra: Key to the

mechanics of Schrödinger and Heisenberg which has been so effective in explain-

Structure of the Atom

ing the structure of matter. The discovery of the electron in the 1890s, with which

27–12 The Bohr Model

we begin this Chapter, might be said to mark the beginning of modern physics,

de Broglie’s Hypothesis

and is a sort of precursor to the quantum theory.

Applied to Atoms

27–1 Discovery and Properties of

the Electron

Screens Toward the end of the nineteenth century, studies were being done on the Cathode

discharge of electricity through rarefied gases. One apparatus, diagrammed in Fig. 27–1, was a glass tube fitted with electrodes and evacuated so only a small amount of gas remained inside. When a very high voltage was applied to the

Glow

Anode

electrodes, a dark space seemed to extend outward from the cathode (negative

– High +

electrode) toward the opposite end of the tube; and that far end of the tube would voltage

glow. If one or more screens containing a small hole were inserted as shown, FIGURE 27–1 Discharge tube. In

the glow was restricted to a tiny spot on the end of the tube. It seemed as though some models, one of the screens is

something being emitted by the cathode traveled across to the opposite end of the the anode (positive plate).

tube. These “somethings” were named cathode rays.

There was much discussion at the time about what these rays might be. Some scientists thought they might resemble light. But the observation that the bright spot at the end of the tube could be deflected to one side by an electric or magnetic field suggested that cathode rays were charged particles; and the direction of the deflection was consistent with a negative charge. Furthermore, if the tube con- tained certain types of rarefied gas, the path of the cathode rays was made visible by a slight glow.

Estimates of the charge e of the cathode-ray particles, as well as of their charge-to-mass ratio e兾m, had been made by 1897. But in that year, J. J. Thomson (1856–1940) was able to measure e兾m directly, using the apparatus shown in Fig. 27–2. Cathode rays are accelerated by a high voltage and then pass between

a pair of parallel plates built into the tube. Another voltage applied to the B

parallel plates produces an electric field B E , and a pair of coils produces a magnetic field B . If E = B = 0, the cathode rays follow path b in Fig. 27–2.

Anode

FIGURE 27–2 Cathode rays

deflected by electric and magnetic

b fields. (See also Section 17–11 on

the CRT.) –

High voltage

Electric field

plates

Coils to produce magnetic field

When only the electric field is present, say with the upper plate positive, the cathode rays are deflected upward as in path a in Fig. 27–2. If only a magnetic field exists, say inward, the rays are deflected downward along path c. These observations are just what is expected for a negatively charged particle. The force on the rays due to the magnetic field is

F = evB, where e is the charge and v is the velocity of the cathode rays (Eq. 20–4). In the absence of an electric field, the rays are bent into a curved path, and applying Newton’s second law

F = ma

with a = centripetal acceleration gives v 2

evB = m , r

and thus

Br

The radius of curvature r can be measured and so can B. The velocity v can be found by applying an electric field in addition to the magnetic field. The electric

772 CHAPTER 27 Early Quantum Theory and Models of the Atom 772 CHAPTER 27 Early Quantum Theory and Models of the Atom

F = eE, is

balanced by the downward force due to the magnetic field,

F = evB. We equate

the two forces, eE = evB, and find

E v=

B Combining this with the above equation we have

The quantities on the right side can all be measured, and although e and m could not be determined separately, the ratio e兾m could be determined. The accepted

value today is 11 e兾m = 1.76 * 10 C兾kg. Cathode rays soon came to be called

electrons.

Discovery in Science

The “discovery” of the electron, like many others in science, is not quite so obvious as discovering gold or oil. Should the discovery of the electron be credited to the person who first saw a glow in the tube? Or to the person who first called them cathode rays? Perhaps neither one, for they had no conception of the electron as we know it today. In fact, the credit for the discovery is generally given to Thomson, but not because he was the first to see the glow in the tube. Rather it is because he believed that this phenomenon was due to tiny negatively charged particles and made careful measurements on them. Furthermore he argued that these particles were constituents of atoms, and not ions or atoms themselves as many thought, and he developed an electron theory of matter. His view is close to what we accept today, and this is why Thomson is credited with the “discovery.” Note, however, that neither he nor anyone else ever actually saw an electron itself. We discuss this briefly, for it illustrates the fact that discovery in science is not always a clear-cut matter. In fact some philosophers of science think the word “discovery” is often not appropriate, such as in this case.

Electron Charge Measurement

Atomizer Thomson believed that an electron was not an atom, but rather a constituent,

or part, of an atom. Convincing evidence for this came soon with the determin- ation of the charge and the mass of the cathode rays. Thomson’s student

+ J. S. Townsend made the first direct (but rough) measurements of e in 1897. But + ⫹ +

⫹ ⫹⫹ ⫹⫹⫹⫹⫹⫹⫹ + ⫹⫹ it was the more refined oil-drop experiment of Robert A. Millikan (1868–1953) Droplets

that yielded a precise value for the charge on the electron and showed that charge

– comes in discrete amounts. In this experiment, tiny droplets of mineral oil carrying – ⫺ –

Telescope

an electric charge were allowed to fall under gravity between two parallel plates, ⫺⫺ Fig. 27–3. The electric field E between the plates was adjusted until the drop was FIGURE 27–3 Millikan’s oil-drop

suspended in midair. The downward pull of gravity, mg, was then just balanced by experiment. the upward force due to the electric field. Thus qE = mg so the charge q = mg兾E. The mass of the droplet was determined by measuring its terminal velocity in the absence of the electric field. Often the droplet was charged negatively, but some- times it was positive, suggesting that the droplet had acquired or lost electrons (by friction, leaving the atomizer). Millikan’s painstaking observations and analysis pre- sented convincing evidence that any charge was an integral multiple of a smallest charge, e, that was ascribed to the electron, and that the value of e was

1.6 * 10 –19 C.

This value of e, combined with the measurement of e兾m, gives the mass of the

electron to be 11 A1.6 * 10 –19 C B兾A1.76 * 10 C兾kg B=

9.1 * 10 –31 kg. This mass is

less than a thousandth the mass of the smallest atom, and thus confirmed the idea that the electron is only a part of an atom. The accepted value today for the mass of the electron is

m e = 9.11 * 10 –31 kg. The experimental result that any charge is an integral multiple of e means that

electric charge is quantized (exists only in discrete amounts).

SECTION 27–1 773

27–2 Blackbody Radiation;

Planck’s Quantum Hypothesis

Blackbody Radiation

One of the observations that was unexplained at the end of the nineteenth cen- tury was the spectrum of light emitted by hot objects. We saw in Section 14–8 that all objects emit radiation whose total intensity is proportional to the fourth

power of the Kelvin (absolute) temperature 4 AT B. At normal temperatures (L 300 K), we are not aware of this electromagnetic radiation because of its low Frequency (Hz)

intensity. At higher temperatures, there is sufficient infrared radiation that we

can feel heat if we are close to the object. At still higher temperatures (on the order of 1000 K), objects actually glow, such as a red-hot electric stove burner or the heating element in a toaster. At temperatures above 2000 K, objects glow with

a yellow or whitish color, such as white-hot iron and the filament of a lightbulb. The light emitted contains a continuous range of wavelengths or frequencies, and the spectrum is a plot of intensity vs. wavelength or frequency. As the temperature

increases, the electromagnetic radiation emitted by objects not only increases in Intensity

6000 K

total intensity but has its peak intensity at higher and higher frequencies.

4500 K

The spectrum of light emitted by a hot dense object is shown in Fig. 27–4 for an idealized blackbody. A blackbody is a body that, when cool, would absorb all

3000 K

the radiation falling on it (and so would appear black under reflection when illuminated by other sources). The radiation such an idealized blackbody would

0 UV 1000 IR 2000

emit when hot and luminous, called blackbody radiation (though not necessarily Visible

Wavelength (nm) black in color), approximates that from many real objects. The 6000-K curve in FIGURE 27–4 Measured spectra of

Fig. 27–4, corresponding to the temperature of the surface of the Sun, peaks in wavelengths and frequencies

the visible part of the spectrum. For lower temperatures, the total intensity drops emitted by a blackbody at three

considerably and the peak occurs at longer wavelengths (or lower frequencies). different temperatures.

This is why objects glow with a red color at around 1000 K. It is found experimen- tally that the wavelength at the peak of the spectrum, l P , is related to the Kelvin temperature T by

l T = 2.90 * 10 –3 P m ⭈ K. (27;2)

This is known as Wien’s law.

EXAMPLE 27;1 The Sun’s surface temperature. Estimate the tempera- ture of the surface of our Sun, given that the Sun emits light whose peak intensity

occurs in the visible spectrum at around 500 nm.

APPROACH We assume the Sun acts as a blackbody, and use l P = 500 nm in

Wien’s law (Eq. 27–2). SOLUTION Wien’s law gives

2.90 * 10 –3 m⭈K =

2.90 * 10 –3 m⭈K

T=

L 6000 K.

500 * 10 –9 m EXAMPLE 27;2 Star color. Suppose a star has a surface temperature of

32,500 K. What color would this star appear? APPROACH We assume the star emits radiation as a blackbody, and solve for

l P in Wien’s law, Eq. 27–2. SOLUTION From Wien’s law we have

2.90 * 10 –3 m⭈K 2.90 * 10 = –3 = m⭈K

= 89.2 nm.

3.25 * 10 4 K The peak is in the UV range of the spectrum, and will be way to the left in Fig. 27–4. In the visible region, the curve will be descending, so the shortest visible

wavelengths will be strongest. Hence the star will appear bluish (or blue-white). NOTE This example helps us to understand why stars have different colors

(reddish for the coolest stars; orangish, yellow, white, bluish for “hotter” stars.)

CHAPTER 27 EXERCISE A What is the color of an object at 4000 K?

Planck’s Quantum Hypothesis

In the year 1900, Max Planck (1858–1947) proposed a theory that was able to reproduce the graphs of Fig. 27–4. His theory, still accepted today, made a new and radical assumption: that the energy of the oscillations of atoms within molecules cannot have just any value; instead each has energy which is a multiple of a mini- mum value related to the frequency of oscillation by

E = hf. Here h is a new constant, now called Planck’s constant, whose value was

estimated by Planck by fitting his formula for the blackbody radiation curve to experiment. The value accepted today is

h = 6.626 * 10 –34 J ⭈ s. Planck’s assumption suggests that the energy of any molecular vibration could be

only a whole number multiple of hf :

n = 1, 2, 3, p,

E = nhf,

where n is called a quantum number (“quantum” means “discrete amount” as opposed to “continuous”). This idea is often called Planck’s quantum hypothesis, although little attention was brought to this point at the time. In fact, it appears that Planck considered it more as a mathematical device to get the “right answer” rather than as an important discovery. Planck himself continued to seek a classical explanation for the introduction of h. The recognition that this was an important

(a) and radical innovation did not come until later, after about 1905 when others,

particularly Einstein, entered the field. The quantum hypothesis, Eq. 27–3, states that the energy of an oscillator can be

E = hf, or 2hf, or 3hf, and so on, but there cannot be vibrations with energies between these values. That is, energy would not be a continuous quan- tity as had been believed for centuries; rather it is quantized—it exists only in discrete amounts. The smallest amount of energy possible (hf) is called the quantum of energy. Recall from Chapter 11 that the energy of an oscillation is proportional to the amplitude squared. Another way of expressing the quantum

(b) hypothesis is that not just any amplitude of vibration is possible. The possible FIGURE 27–5 Ramp versus stair values for the amplitude are related to the frequency f .

analogy. (a) On a ramp, a box can have

A simple analogy may help. Compare a ramp, on which a box can be placed continuous values of potential energy. at any height, to a flight of stairs on which the box can have only certain discrete (b) But on stairs, the box can have amounts of potential energy, as shown in Fig. 27–5.

only discrete (quantized) values of energy.

27–3 Photon Theory of Light and

the Photoelectric Effect

In 1905, the same year that he introduced the special theory of relativity, Einstein made a bold extension of the quantum idea by proposing a new theory of light. Planck’s work had suggested that the vibrational energy of molecules in a radiat- ing object is quantized with energy

E = nhf, where n is an integer and f is the

frequency of molecular vibration. Einstein argued that when light is emitted by a molecular oscillator, the molecule’s vibrational energy of nhf must decrease by an amount hf (or by 2hf, etc.) to another integer times hf, such as (n - 1)hf. Then to conserve energy, the light ought to be emitted in packets, or quanta, each with an energy

E = hf, (27;4) Photon energy where f is here the frequency of the emitted light. Again h is Planck’s constant.

Because all light ultimately comes from a radiating source, this idea suggests that light is transmitted as tiny particles , or photons as they are now called, as well as via the waves predicted by Maxwell’s electromagnetic theory. The photon theory of light was also a radical departure from classical ideas. Einstein proposed a test of the quantum theory of light: quantitative measurements on the photoelectric effect.

SECTION 27–3 Photon Theory of Light and the Photoelectric Effect 775

Light When light shines on a metal surface, electrons are found to be emitted from source

the surface. This effect is called the photoelectric effect and it occurs in many materials, but is most easily observed with metals. It can be observed using the apparatus shown in Fig. 27–6. A metal plate P and a smaller electrode C are placed inside an evacuated glass tube, called a photocell. The two electrodes are

Light connected to an ammeter and a source of emf, as shown. When the photocell is in the dark, the ammeter reads zero. But when light of sufficiently high frequency illuminates the plate, the ammeter indicates a current flowing in the circuit. We

P explain completion of the circuit by imagining that electrons, ejected from the

plate by the impinging light, flow across the tube from the plate to the “collector” C as indicated in Fig. 27–6.

Photocell That electrons should be emitted when light shines on a metal is consistent

A with the electromagnetic (EM) wave theory of light: the electric field of an EM wave could exert a force on electrons in the metal and eject some of them.

V Einstein pointed out, however, that the wave theory and the photon theory of

light give very different predictions on the details of the photoelectric effect. For FIGURE 27–6 The photoelectric

example, one thing that can be measured with the apparatus of Fig. 27–6 is the effect.

maximum kinetic energy Ake max B of the emitted electrons. This can be done by using a variable voltage source and reversing the terminals so that electrode C is negative and P is positive. The electrons emitted from P will be repelled by the negative electrode, but if this reverse voltage is small enough, the fastest electrons will still reach C and there will be a current in the circuit. If the reversed voltage is increased, a point is reached where the current reaches zero—no electrons have sufficient kinetic energy to reach C. This is called the stopping potential, or

stopping voltage , V 0 , and from its measurement, ke max can be determined using conservation of energy (loss of kinetic energy = gain in potential energy):

ke max = eV 0 .

Now let us examine the details of the photoelectric effect from the point of view of the wave theory versus Einstein’s particle theory.

First the wave theory, assuming monochromatic light. The two important properties of a light wave are its intensity and its frequency (or wavelength). When these two quantities are varied, the wave theory makes the following predictions:

Wave

1. If the light intensity is increased, the number of electrons ejected and their maximum kinetic energy should be increased because the higher intensity

theory

means a greater electric field amplitude, and the greater electric field should eject electrons with higher speed.

predictions

2. The frequency of the light should not affect the kinetic energy of the ejected electrons. Only the intensity should affect ke max .

The photon theory makes completely different predictions. First we note that in a monochromatic beam, all photons have the same energy (= hf). Increasing the intensity of the light beam means increasing the number of photons in the beam, but does not affect the energy of each photon as long as the frequency is not changed. According to Einstein’s theory, an electron is ejected from the metal by a collision with a single photon. In the process, all the photon energy is trans- ferred to the electron and the photon ceases to exist. Since electrons are held in

the metal by attractive forces, some minimum energy W 0 is required just to get an electron out through the surface. W 0 is called the work function, and is a few electron volts A1 eV = 1.6 * 10 –19 J B for most metals. If the frequency f of the incoming light is so low that hf is less than W 0 , then the photons will not have

enough energy to eject any electrons at all. If

hf 7 W 0 , then electrons will be ejected and energy will be conserved in the process. That is, the input energy (of the photon), hf, will equal the outgoing kinetic energy ke of the electron plus the energy required to get it out of the metal, W:

hf = ke + W.

(27;5a)

The least tightly held electrons will be emitted with the most kinetic energy Ake max B,

776 CHAPTER 27 Early Quantum Theory and Models of the Atom 776 CHAPTER 27 Early Quantum Theory and Models of the Atom

becomes ke max :

hf = ke max +W 0 .

[least bound electrons] (27;5b)

Many electrons will require more energy than the bare minimum AW 0 B to get out

of the metal, and thus the kinetic energy of such electrons will be less than the maximum.

From these considerations, the photon theory makes the following predictions:

1. An increase in intensity of the light beam means more photons are incident, so more electrons will be ejected; but since the energy of each photon is not Photon changed, the maximum kinetic energy of electrons is not changed by an increase in intensity.

2. If the frequency of the light is increased, the maximum kinetic energy of the theory electrons increases linearly, according to Eq. 27–5b. That is,

This relationship is plotted in Fig. 27–7.

3. If the frequency f is less than the “cutoff” frequency f 0 , where hf 0 =W 0 , no

electrons will be ejected, no matter how great the intensity of the light.

max

These predictions of the photon theory are very different from the predictions

KE

of the wave theory. In 1913–1914, careful experiments were carried out by R. A.

of electrons

Millikan. The results were fully in agreement with Einstein’s photon theory. One other aspect of the photoelectric effect also confirmed the photon

f 0 Frequency of light f theory. If extremely low light intensity is used, the wave theory predicts a time FIGURE 27–7 Photoelectric effect:

delay before electron emission so that an electron can absorb enough energy to the maximum kinetic energy of exceed the work function. The photon theory predicts no such delay—it only ejected electrons increases linearly takes one photon (if its frequency is high enough) to eject an electron—and with the frequency of incident light. experiments showed no delay. This too confirmed Einstein’s photon theory.

No electrons are emitted if f 6 f 0 . EXAMPLE 27;3 Photon energy. Calculate the energy of a photon of blue

light, l= 450 nm in air (or vacuum). APPROACH The photon has energy

E = hf (Eq. 27–4) where f = c兾l

(Eq. 22–4). SOLUTION Since f = c兾l, we have

hc 8 A6.63 * 10 J⭈s BA3.00 * 10 m兾s

A4.5 * 10 m B

or A4.4 * 10 –19 J B 兾 A1.60 * 10 –19 J兾eV B = 2.8 eV. (See definition of eV in

1 eV = 1.60 * 10 Section 17–4, –19 J. )

EXAMPLE 27;4 ESTIMATE Photons from a lightbulb. Estimate how many visible light photons a 100-W lightbulb emits per second. Assume the bulb has a typical efficiency of about 3% (that is, 97% of the energy goes to heat).

APPROACH Let’s assume an average wavelength in the middle of the visible

spectrum, lL 500 nm. The energy of each photon is

E = hf = hc兾l. Only

3% of the 100-W power is emitted as visible light, or 3 W=3 J兾s. The number of photons emitted per second equals the light output of 3 J兾s divided by

the energy of each photon.

SOLUTION The energy emitted in one second (= 3 J) is

E = Nhf where N is

the number of photons emitted per second and

f = c兾l. Hence

(3 J) A500 * 10 E –9 = El

hf hc A6.63 * 10 8 –34 J⭈s BA3.00 * 10 m兾s B

per second, or almost 19 10 photons emitted per second, an enormous number.

SECTION 27–3 Photon Theory of Light and the Photoelectric Effect 777

EXERCISE B

A beam contains infrared light of a single wavelength, 1000 nm, and monochromatic UV at 100 nm, both of the same intensity. Are there more 100-nm photons or more 1000-nm photons?

EXAMPLE 27;5 Photoelectron speed and energy. What is the kinetic energy and the speed of an electron ejected from a sodium surface whose work

function is W 0 = 2.28 eV when illuminated by light of wavelength (a) 410 nm,

(b) 550 nm? APPROACH We first find the energy of the photons (E = hf = hc兾l). If the

energy is greater than W 0 , then electrons will be ejected with varying amounts

of ke, with a maximum of ke max = hf - W 0 . SOLUTION (a) For l= 410 nm,

hc

4.85 * 10 –19 J or 3.03 eV.

hf =

The maximum kinetic energy an electron can have is given by Eq. 27–5b, ke max = 3.03 eV - 2.28 eV = 0.75 eV, or (0.75 eV)(1.60 * 10 –19 J兾eV) =

J. 1 Since where mv 2 ke =

2 m = 9.1 * 10 –31 kg,

2ke

5.1 * 10 max 5 m兾s. B

Most ejected electrons will have less ke and less speed than these maximum values.

(b) For l= 550 nm, hf = hc兾l = 3.61 * 10 –19 J = 2.26 eV. Since this photon energy is less than the work function, no electrons are ejected.

NOTE In (a) we used the nonrelativistic equation for kinetic energy. If v had turned out to be more than about 0.1c, our calculation would have been inaccurate by more than a percent or so, and we would probably prefer to redo it using the relativistic form (Eq. 26–5).

EXERCISE C Determine the lowest frequency and the longest wavelength needed to emit electrons from sodium.

By converting units, we can show that the energy of a photon in electron volts, when given the wavelength in nm, is l

1.240 * 10 3 eV ⭈ nm

E (eV) = .

[photon energy in eV]

l (nm)

Applications of the Photoelectric Effect

The photoelectric effect, besides playing an important historical role in confirm- ing the photon theory of light, also has many practical applications. Burglar alarms

FIGURE 27–8 Optical sound track and automatic doors often make use of the photocell circuit of Fig. 27–6. When on movie film. In the projector, light

a person interrupts the beam of light, the sudden drop in current in the circuit from a small source (different from

that for the picture) passes through activates a switch—often a solenoid—which operates a bell or opens the door.

the sound track on the moving film. UV or IR light is sometimes used in burglar alarms because of its invisibility. Many Picture

smoke detectors use the photoelectric effect to detect tiny amounts of smoke Sound track

that interrupt the flow of light and so alter the electric current. Photographic light meters use this circuit as well. Photocells are used in many other devices, such as

Photocell absorption spectrophotometers, to measure light intensity. One type of film sound track is a variably shaded narrow section at the side of the film, Fig. 27–8. Light

passing through the film is thus “modulated,” and the output electrical signal of the photocell detector follows the frequencies on the sound track. For many

Small light

applications today, the vacuum-tube photocell of Fig. 27–6 has been replaced by a source

semiconductor device known as a photodiode (Section 29–9). In these semicon- ductors, the absorption of a photon liberates a bound electron so it can move freely, which changes the conductivity of the material and the current through a photodiode is altered.

778 CHAPTER 27 Early Quantum Theory and Models of the Atom

27–4 Energy, Mass, and

Momentum of a Photon

We have just seen (Eq. 27–4) that the total energy of a single photon is given by

E = hf. Because a photon always travels at the speed of light, it is truly a rela- tivistic particle. Thus we must use relativistic formulas for dealing with its mass, energy, and momentum. The momentum of any particle of mass m is given by p = mv兾

2 兾c 31 - v 2 . Since v=c for a photon, the denominator is zero. To avoid having an infinite momentum, we conclude that the photon’s mass must be

zero: m = 0. This makes sense too because a photon can never be at rest (it always moves at the speed of light). A photon’s kinetic energy is its total energy:

ke = E = hf.

[photon]

The momentum of a photon can be obtained from the relativistic formula

E 2 (Eq. 26–9) 2 =p c 2 2 c 4 where we set

c +m 2 m = 0, so E 2 =p 2 or

E p=

c CAUTION Momentum of photon is not mv Since

[photon]

E = hf for a photon, its momentum is related to its wavelength by

EXAMPLE 27;6 ESTIMATE Photon momentum and force. Suppose the

10 19 photons emitted per second from the 100-W lightbulb in Example 27–4 were all focused onto a piece of black paper and absorbed. (a) Calculate the momentum of one photon and (b) estimate the force all these photons could exert on the paper.

APPROACH Each photon’s momentum is obtained from Eq. 27–6, p = h兾l. Next, each absorbed photon’s momentum changes from p = h兾l to zero. We use Newton’s second law,

F = ¢p兾¢ t , to get the force. Let l= 500 nm.

SOLUTION (a) Each photon has a momentum

h –34

6.63 * 10 J⭈s =

p=

1.3 * 10 –27 kg ⭈ m兾s.

500 * 10 m

(b) Using Newton’s second law for N = 10 19 photons (Example 27–4) whose

momentum changes from h兾l to 0, we obtain ¢p

F= = Nh兾l - 0 =

19 L A10 s –1 BA10 –27 kg ⭈ m兾s B L 10 –8 N.

NOTE This is a tiny force, but we can see that a very strong light source could exert a measurable force, and near the Sun or a star the force due to photons in electromagnetic radiation could be considerable. See Section 22–6.

EXAMPLE 27;7 Photosynthesis. In photosynthesis, pigments such as

PHYSICS APPLIED

Photosynthesis carbohydrate. About nine photons are needed to transform one molecule of CO 2

chlorophyll in plants capture the energy of sunlight to change CO 2 to useful

to carbohydrate and O 2 . Assuming light of wavelength l= 670 nm (chlorophyll

absorbs most strongly in the range 650 nm to 700 nm), how efficient is the photosynthetic process? The reverse chemical reaction releases an energy of

4.9 eV兾molecule of CO 2 , so 4.9 eV is needed to transform CO 2 to carbohydrate.

APPROACH The efficiency is the minimum energy required (4.9 eV) divided by the actual energy absorbed, nine times the energy (hf) of one photon.

SOLUTION The energy of nine photons, each of energy

hf = hc兾l , is

A6.63 * 10 8 –34 J⭈s BA3.00 * 10 m兾s B兾A6.7 * 10 –7 m B = 2.7 * 10 –18 J or 17 eV.

Thus the process is about (4.9 eV兾17 eV) = 29% efficient.

SECTION 27–4 Energy, Mass, and Momentum of a Photon 779

* 27–5 Compton Effect

BEFORE AFTER Besides the photoelectric effect, a number of other experiments were carried out COLLISION

COLLISION in the early twentieth century which also supported the photon theory. One of y Scattered

these was the Compton effect (1923) named after its discoverer, A. H. Compton photon ( λ ')

(1892–1962). Compton aimed short-wavelength light (actually X-rays) at various Incident photon ( ) λ

materials, and detected light scattered at various angles. He found that the scattered

light had a slightly longer wavelength than did the incident light, and therefore a

Electron slightly lower frequency indicating a loss of energy. He explained this result on the at rest

basis of the photon theory as incident photons colliding with electrons of the initially

e− material, Fig. 27–9. Using Eq. 27–6 for momentum of a photon, Compton applied the laws of conservation of momentum and energy to the collision of Fig. 27–9

FIGURE 27–9 The Compton effect. and derived the following equation for the wavelength of the scattered photons: A single photon of wavelength l

strikes an electron in some material,

l¿ = l +

(1 - cos f),

m e c (27;7)

knocking it out of its atom. The

scattered photon has less energy

e c, which has the dimen- electron) and hence has a longer

where m e is the mass of the electron. (The quantity h兾m

(some energy is given to the

sions of length, is called the Compton wavelength of the electron.) We see that wavelength (shown l¿ exaggerated).

the predicted wavelength of scattered photons depends on the angle at which f Experiments found scattered X-rays

they are detected. Compton’s measurements of 1923 were consistent with this of just the wavelengths predicted by

formula. The wave theory of light predicts no such shift: an incoming electro- conservation of energy and

magnetic wave of frequency f should set electrons into oscillation at frequency f; momentum using the photon model.

and such oscillating electrons would reemit EM waves of this same frequency f (Section 22–2), which would not change with angle (f). Hence the Compton effect adds to the firm experimental foundation for the photon theory of light.

EXERCISE D When a photon scatters off an electron by the Compton effect, which of the following increases: its energy, frequency, wavelength?

EXAMPLE 27;8 X-ray scattering. X-rays of wavelength 0.140 nm are scattered from a very thin slice of carbon. What will be the wavelengths of X-rays

scattered at (a) 0°, (b) 90°, (c) 180°? APPROACH This is an example of the Compton effect, and we use Eq. 27–7 to

find the wavelengths. SOLUTION (a) For f= 0°, cos f = 1 and 1 - cos f = 0. Then Eq. 27–7

gives l¿ = l = 0.140 nm. This makes sense since for f= 0°, there really isn’t any collision as the photon goes straight through without interacting.

(b) For f= 90°, cos f = 0, and 1 - cos f = 1. So

h 6.63 * 10 –34 J⭈s

l¿ = l +

= 0.140 nm +

m e c –31 kg A9.11 * 10 8 BA3.00 * 10 m兾s B = 0.140 nm + 2.4 * 10 –12 m = 0.142 nm; that is, the wavelength is longer by one Compton wavelength (= h兾m e c 2

= 0.0024 nm for an electron).

(c) For f= 180°, which means the photon is scattered backward, returning in the direction from which it came (a direct “head-on” collision), cos f = –1, and 1 - cos f = 2. So

l¿ = l + 2 = 0.140 nm + 2(0.0024 nm) = 0.145 nm.

NOTE The maximum shift in wavelength occurs for backward scattering, and it is twice the Compton wavelength.

PHYSICS APPLIED

The Compton effect has been used to diagnose bone disease such as osteoporo- Measuring bone density sis. Gamma rays, which are photons of even shorter wavelength than X-rays, coming from a radioactive source are scattered off bone material. The total intensity of the scattered radiation is proportional to the density of electrons, which is in turn proportional to the bone density. A low bone density may indicate osteoporosis.

780 CHAPTER 27 Early Quantum Theory and Models of the Atom

27–6 Photon Interactions; Pair Production

When a photon passes through matter, it interacts with the atoms and electrons. There are four important types of interactions that a photon can undergo:

1. The photoelectric effect: A photon may knock an electron out of an atom and in the process the photon disappears.

2. The photon may knock an atomic electron to a higher energy state in the atom if its energy is not sufficient to knock the electron out altogether. In this process

e ⫹ the photon also disappears, and all its energy is given to the atom. Such an

atom is then said to be in an excited state, and we shall discuss it more later.

3. The photon can be scattered from an electron (or a nucleus) and in the Photon process lose some energy; this is the Compton effect (Fig. 27–9). But notice

that the photon is not slowed down. It still travels with speed c, but its + frequency will be lower because it has lost some energy.

Nucleus

e 4. Pair production: A photon can actually create matter, such as the production − of an electron and a positron, Fig. 27–10. (A positron has the same mass as FIGURE 27–10 Pair production:

an electron, but the opposite charge, ±e. ) a photon disappears and produces an In process 4, pair production, the photon disappears in the process of creating electron and a positron. the electron–positron pair. This is an example of mass being created from pure

energy, and it occurs in accord with Einstein’s equation

E = mc 2 . Notice that a

photon cannot create an electron alone since electric charge would not then be conserved. The inverse of pair production also occurs: if a positron comes close to an electron, the two quickly annihilate each other and their energy, including their mass, appears as electromagnetic energy of photons. Because positrons are not as plentiful in nature as electrons, they usually do not last long.

Electron–positron annihilation is the basis for the type of medical imaging known as PET, as discussed in Section 31–8.

EXAMPLE 27;9 Pair production. (a) What is the minimum energy of a photon that can produce an electron–positron pair? (b) What is this photon’s wavelength?

APPROACH The minimum photon energy E equals the rest energy

Amc 2 B of

the two particles created, via Einstein’s famous equation

E = mc 2 (Eq. 26–7).

There is no energy left over, so the particles produced will have zero kinetic energy. The wavelength is l= c兾f where

E = hf for the original photon.

SOLUTION (a) Because

E = mc 2 , and the mass created is equal to two electron

masses, the photon must have energy E=2

A9.11 * 10 8 –31 kg

2 BA3.00 * 10 m兾s B =

1.64 * 10 –13 J = 1.02 MeV

(1 MeV = 10 6 eV = 1.60 * 10 –13 J). A photon with less energy cannot undergo

pair production. (b) Since

E = hf = hc兾l, the wavelength of a 1.02-MeV photon is

hc 8 A6.63 * 10 –34 J⭈s BA3.00 * 10 m兾s B

E A1.64 * 10 J B

which is 0.0012 nm. Such photons are in the gamma-ray (or very short X-ray) region of the electromagnetic spectrum (Fig. 22–8). NOTE Photons of higher energy (shorter wavelength) can also create an electron– positron pair, with the excess energy becoming kinetic energy of the particles.

Pair production cannot occur in empty space, for momentum could not be con- served. In Example 27–9, for instance, energy is conserved, but only enough energy was provided to create the electron–positron pair at rest and thus with zero momen- tum, which could not equal the initial momentum of the photon. Indeed, it can be shown that at any energy, an additional massive object, such as an atomic nucleus (Fig. 27–10), must take part in the interaction to carry off some of the momentum.

SECTION 27–6 Photon Interactions; Pair Production 781

27–7 Wave–Particle Duality; the

Principle of Complementarity

The photoelectric effect, the Compton effect, and other experiments have placed the particle theory of light on a firm experimental basis. But what about the classic experiments of Young and others (Chapter 24) on interference and diffraction which showed that the wave theory of light also rests on a firm experimental basis?

We seem to be in a dilemma. Some experiments indicate that light behaves like

a wave; others indicate that it behaves like a stream of particles. These two theories seem to be incompatible, but both have been shown to have validity. Physicists finally came to the conclusion that this duality of light must be accepted as a fact of life. It is referred to as the wave;particle duality. Apparently, light is a more complex phenomenon than just a simple wave or a simple beam of particles.

To clarify the situation, the great Danish physicist Niels Bohr (1885–1962, Fig. 27–11) proposed his famous principle of complementarity. It states that to understand an experiment, sometimes we find an explanation using wave theory and sometimes using particle theory. Yet we must be aware of both the wave and

FIGURE 27–11 Niels Bohr (right), particle aspects of light if we are to have a full understanding of light. Therefore walking with Enrico Fermi along the

these two aspects of light complement one another.

Appian Way outside Rome. This

photo shows one important way It is not easy to “visualize” this duality. We cannot readily picture a combina- physics is done.

tion of wave and particle. Instead, we must recognize that the two aspects of light are different “faces” that light shows to experimenters.

Part of the difficulty stems from how we think. Visual pictures (or models) in our minds are based on what we see in the everyday world. We apply the concepts of waves and particles to light because in the macroscopic world we see that energy is transferred from place to place by these two methods. We cannot see directly whether light is a wave or particle, so we do indirect experiments. To explain the experiments, we apply the models of waves or of particles to the nature of light. But these are abstractions of the human mind. When we try to conceive of what light really “is,” we insist on a visual picture. Yet there is no

reason why light should conform to these models (or visual images) taken from Not correct to say light is a wave and/or the macroscopic world. The “true” nature of light—if that means anything—is

CAUTION

a particle. Light can act like a wave or not possible to visualize. The best we can do is recognize that our knowledge is

like a particle limited to the indirect experiments, and that in terms of everyday language and

images, light reveals both wave and particle properties.

E = hf itself links the particle and wave properties of a light beam. In this equation, E refers to the energy of a particle; and on the other side of the equation, we have the frequency f of the corresponding wave.

It is worth noting that Einstein’s equation

27–8 Wave Nature of Matter

In 1923, Louis de Broglie (1892–1987) extended the idea of the wave–particle duality. He appreciated the symmetry in nature, and argued that if light some- times behaves like a wave and sometimes like a particle, then perhaps those things in nature thought to be particles—such as electrons and other material objects— might also have wave properties. De Broglie proposed that the wavelength of a material particle would be related to its momentum in the same way as for a photon, Eq. 27–6, p = h兾l. That is, for a particle having linear momentum p = mv, the wavelength is given by l

de Broglie wavelength

l=

and is valid classically ( p = mv for vVc ) and relativistically Ap = gmv = mv兾

31 - v 2 兾c 2 B. This is sometimes called the de Broglie wavelength of a

particle.

782 CHAPTER 27 Early Quantum Theory and Models of the Atom

EXAMPLE 27;10 Wavelength of a ball. Calculate the de Broglie wavelength

of a 0.20-kg ball moving with a speed of 15 m兾s.

APPROACH We use Eq. 27–8.

= h = –34

h A6.6 * 10 J⭈s B

SOLUTION l=

= 2.2 * 10 –34 m.

mv

(0.20 kg)(15 m兾s)

Ordinary objects, such as the ball of Example 27–10, have unimaginably small

wavelengths. Even if the speed is extremely small, say 10 –4 m兾s, the wavelength would be about 10 –29 m. Indeed, the wavelength of any ordinary object is much

too small to be measured and detected. The problem is that the properties of waves, such as interference and diffraction, are significant only when the size of objects or slits is not much larger than the wavelength. And there are no known

objects or slits to diffract waves only 10 –30 m long, so the wave properties of

ordinary objects go undetected. But tiny elementary particles, such as electrons, are another matter. Since the mass m appears in the denominator of Eq. 27–8, a very small mass should have a much larger wavelength.

EXAMPLE 27;11 Wavelength of an electron. Determine the wavelength of an electron that has been accelerated through a potential difference of 100 V.

APPROACH If the kinetic energy is much less than the rest energy, we can use the classical formula,

ke = 1 mv 2

2 (see end of Section 26–9). For an electron,

mc 2 = 0.511 MeV. We then apply conservation of energy: the kinetic energy acquired by the electron equals its loss in potential energy. After solving for v, we use Eq. 27–8 to find the de Broglie wavelength.

SOLUTION The gain in kinetic energy equals the loss in potential energy:

6 ke = eV,

ke = 100

ke兾mc

¢pe = eV - 0. Thus so eV. The ratio 2 =

100 eV兾

A0.511 * 10 eV B L 10 –4 , so relativity is not needed. Thus

1 mv 2 = eV

2 and

2 eV –19 (2) A1.6 * 10 C B(100 V) = =

5.9 * 10 6 v=B m兾s.

h A6.63 * 10 –34 J⭈s B

kg BA5.9 * 10 6 m兾s B

or 0.12 nm. EXERCISE E As a particle travels faster, does its de Broglie wavelength decrease,

increase, or remain the same?

EXERCISE F

Return to the Chapter-Opening Question, page 771, and answer it again FIGURE 27–12 Diffraction pattern now. Try to explain why you may have answered differently the first time.

of electrons scattered from aluminum foil, as recorded on film.

Electron Diffraction

From Example 27–11, we see that electrons can have wavelengths on the order of 10 –10 m, and even smaller. Although small, this wavelength can be detected:

the spacing of atoms in a crystal is on the order of 10 –10 m and the orderly array

of atoms in a crystal could be used as a type of diffraction grating, as was done earlier for X-rays (see Section 25–11). C. J. Davisson and L. H. Germer per- formed the crucial experiment: they scattered electrons from the surface of a metal crystal and, in early 1927, observed that the electrons were scattered into a pattern of regular peaks. When they interpreted these peaks as a diffraction pattern, the wavelength of the diffracted electron wave was found to be just that predicted by

de Broglie, Eq. 27–8. In the same year, G. P. Thomson (son of J. J. Thomson) used

a different experimental arrangement and also detected diffraction of electrons. (See Fig. 27–12. Compare it to X-ray diffraction, Section 25–11.) Later experiments showed that protons, neutrons, and other particles also have wave properties.

SECTION 27–8 Wave Nature of Matter 783

Thus the wave–particle duality applies to material objects as well as to light. The principle of complementarity applies to matter as well. That is, we must be aware of both the particle and wave aspects in order to have an understanding of matter, including electrons. But again we must recognize that a visual picture of a “wave–particle” is not possible.

PHYSICS APPLIED

EXAMPLE 27;12 Electron diffraction. The wave nature of electrons is mani- Electron diffraction

fested in experiments where an electron beam interacts with the atoms on the surface of a solid, especially crystals. By studying the angular distribution of the

beam electron Incident diffracted electrons, one can indirectly measure the geometrical arrangement of atoms. Assume that the electrons strike perpendicular to the surface of a solid

(see Fig. 27–13), and that their energy is low, ke = 100 eV, so that they interact only with the surface layer of atoms. If the smallest angle at which a diffraction u

maximum occurs is at 24°, what is the separation d between the atoms on the d sin u

surface?

u SOLUTION Treating the electrons as waves, we need to determine the condi-

d tion where the difference in path traveled by the wave diffracted from adjacent atoms is an integer multiple of the de Broglie wavelength, so that constructive

interference occurs. The path length difference is d sin u (Fig. 27–13); so for the smallest value of we must have u

FIGURE 27–13 Example 27–12.

d sin u = l.

The red dots represent atoms in an orderly array in a solid.

However, l is related to the (non-relativistic) kinetic energy ke by

A6.63 * 10 –34 J⭈s B

0.123 nm. 32A9.11 * 10 kg B(100 eV) A1.6 * 10 –19 J兾eV B

The surface inter-atomic spacing is

NOTE Experiments of this type verify both the wave nature of electrons and the orderly array of atoms in crystalline solids.

What Is an Electron?

We might ask ourselves: “What is an electron?” The early experiments of J. J. Thomson (Section 27–1) indicated a glow in a tube, and that glow moved when

a magnetic field was applied. The results of these and other experiments were best interpreted as being caused by tiny negatively charged particles which we now call electrons. No one, however, has actually seen an electron directly. The drawings we sometimes make of electrons as tiny spheres with a negative charge on them are merely convenient pictures (now recognized to be inaccurate). Again we must rely on experimental results, some of which are best interpreted using the particle model and others using the wave model. These models are mere pictures that we use to extrapolate from the macroscopic world to the tiny microscopic world of the atom. And there is no reason to expect that these models somehow reflect the reality of an electron. We thus use a wave or a particle model (whichever works best in a situation) so that we can talk about what is happening. But we should not

be led to believe that an electron is a wave or a particle. Instead we could say that an electron is the set of its properties that we can measure. Bertrand Russell said it well when he wrote that an electron is “a logical construction.”

784 CHAPTER 27 Early Quantum Theory and Models of the Atom

27–9 Electron Microscopes

The idea that electrons have wave properties led to the development of the

PHYSICS APPLIED

electron microscope (EM), which can produce images of much greater magnifi- Electron microscope cation than a light microscope. Figures 27–14 and 27–15 are diagrams of two types,

developed around the middle of the twentieth century: the transmission electron

microscope (TEM), which produces a two-dimensional image, and the scanning

electron microscope (SEM), which produces images with a three-dimensional quality. Electron source Hot filament (source of electrons)

Magnetic lens

High – voltage

and screen

FIGURE 27–14